We consider the quantum switch depicted in Fig. 2.1 [69]. It consists of a QPC with a transmission that is varied in time. The time-dependent Hamiltonian reads
Hˆ0(t) =f(t) ˆHQPC, (3.27)
with f(t) controlling the transmission of the QPC and HˆQPC given by Eq. (2.28).
We rst consider the situation where the QPC initially is closed. We then open it by choosing
f(t) = 1 2 + 1
π arctan t
τ
. (3.28)
Here τ is the opening time of the QPC with τ = 0 corresponding to an abrupt opening. We initialize the disconnected leads at t0 −τ, wheref(t0)≈0.
Figure 3.3a shows the time evolution of the entanglement entropy upon opening the QPC at t = 0. The temperature is zero and no bias voltage is applied between the leads. FortQPC = ¯t, the QPC is fully transmitting and the entanglement entropy is expected to follow the conformal eld theory prediction [122, 106, 109],
S =cln(t/tc) (3.29)
with the central chargec= 1/3and a short-time cutotcon the order of the opening time [69, 70, 71]. Our results clearly follow this prediction, with small oscillations on top of the logarithmic growth if the QPC is abruptly opened. The frequency of the oscillations is given by the distance of the chemical potential to the nearest band edge,ω0 = 2¯t− |µ|, as we have found by systematically varying the occupation of the leads. In addition, the oscillations are smeared out when the opening of the QPC is smooth. For non-perfect transmissions (tQPC <t¯), the entanglement does not follow the logarithmic behavior.
In Fig. 3.3b we turn to the series expansion of the entanglement entropy in terms of the cumulants of the FCS. The gure clearly demonstrates how the exact result is approached as more cumulants are included. In this example, cumulants of very high orders (K ' 30) are needed for the series to converge. However, already with the second cumulant only, the series provides a good approximation of the exact entanglement entropy.
Figure 3.4a shows the time evolution of the Rényi entropies up to order ν = 5.
In this case, we compare the exact results with the Gaussian approximations in Eqs. (3.10) and (3.12). For a fully open QPC (tQPC= ¯t), the Gaussian approximation works very well. The main contribution of the higher cumulants is to smear out the oscillatory behavior. Still, from a measurement of the second cumulant one obtains a good approximation of the entanglement entropy and the Rényi entropies. This only holds for a fully transmitting QPC as illustrated in Fig. 3.4b, showing the series for S(4)in terms of a nite number of cumulants for a QPC with non-unity transmission.
In this case, it is necessary to go beyond the second cumulant before convergence is reached. However, already when the fourth cumulant is included, the series provides a good approximation of the exact Rényi entropy.
Next, we consider the inuence of a nite electronic temperature as well as a nite bias voltage between the leads. The results shown so far were obtained at zero temperature, where the full system is in a pure state and the growth of the entanglement entropy is due to the increasing entanglement between the leads. With
0 10 20 30 40 50
Figure 3.4: Rényi entropies of the quantum switch. a) Rényi entropies for a fully transmitting QPC opened abruptly at t = 0. The thin lines indicate the Gaussian approximation where only the second cumulant is taken into account. b) Rényi entropy of order ν = 4 for a QPC opened abruptly at t = 0 to tQPC = 0.5¯t. The exact result (black line) is shown together with the seriesSK(4) with cumulants up to orderK (colored lines). We have dened the time τ0 =~/¯t.
Figure 3.5: Entanglement entropy at nite temperatures and bias. a) Entanglement entropy for a QPC with a nite electronic temperature (red). Aroundtβ =~/(πkT), the growth of the entanglement entropy changes from logarithmic to being linear in time. The zero temperature result (blue) is shown for comparison together with the linear long-time asymptote (green). b) Entanglement entropy for a nite bias V at zero temperature. For a fully open QPC, the entanglement entropy still grows logarithmically with time. For non-unit transmission, the entanglement entropy becomes linear in time for t & ~/V (dashed vertical line). The unit of time is τ0 =~/t.¯
32
a nite electronic temperature, thermal uctuations come into play together with shot noise due to the applied voltage. This is illustrated in Fig. 3.5a, where we show the time-dependent entanglement entropy for a nite electronic temperature. For nite temperatures, the generalization of the prediction in Eq. (3.29) reads [71, 144]
S = 1 behavior from Eq. (3.29) persists. In contrast, at long times, t tβ, thermal uc-tuations in the Fermi seas cause the entanglement entropy to grow linearly in time.
This crossover is illustrated in Fig. 3.5a, showing the entanglement entropy with and without a nite electronic temperature.
In Fig. 3.5b, we consider a quantum switch with a nite bias voltage between the leads. We take the left and right leads as the source and drain electrodes, respectively, so thatV =µL−µR is the potential dierence. We mimic a nite bias by choosing the initial particle numbers of the leads dierently, giving an approximately constant current during a certain time window (see Sec. 2.1.3). Specically, we choose the initial occupations as N0L/R = (M ±∆N)/2, where ∆N is the surplus of particles in the source electrode. We then have µL = −µR together with the approximate voltage for large tight-binding leads,M 1.
Figure 3.5b shows that the entanglement entropy for a fully transmitting QPC essentially grows logarithmically with time just as in the unbiased case. In contrast, for a QPC with a transmission below unity, electrons in the transport window may reect back on the QPC, generating shot noise. These transport processes can be considered as binomial events, in which each electron is transmitted through the QPC with probabilityDand reected with probability 1−D, as discussed in Ch. 1.
For such binomial processes, the entanglement entropy is expected to follow a linear time dependence at long times following the expression [145]
S(t) =−t
¯
τ [DlogD+ (1−D) log(1−D)], (3.32) where τ¯ =h/V is the mean waiting time between the incoming electrons [54], and the ratio t/¯τ yields the number of transmission attempts after the QPC has been opened. The crossover to the linear behavior is seen in Fig. 3.5b for a biased QPC with a non-unity transmission.
To understand the combined eect of a nite voltage bias and a nite electronic temperature, we consider in Fig. 3.6a the derivative of the entanglement entropy with respect to time for dierent temperatures and voltages as well as dierent transmis-sions of the QPC. Neglecting the quantum noise at zero bias and zero temperature
0.0 0.2 0.4 0.6 0.8 1.0
open closed open closed open closed open
tQPC=¯t, τ=0.1τ0
Figure 3.6: a) Growth of the entanglement entropy at long times. We show dS/dt for t ~/V. Results are shown as functions of the transmission D of the QPC for dierent ratios of the temperature over the bias. Squares are results of our calcula-tions, whereas the solid lines are obtained from Eq. (3.33). b) Periodically driven quantum switch. The transmission of the QPC is controlled by the function F(t) in Eq. (3.35). During each period of duration T = 15τ0 with τ0 = ~/t, the QPC¯ is opened for the time w = 7.5τ0. Since w < tβ = ~/(πkT) ' 32τ0, the loga-rithmic growth dominates within each period. The dashed line shows the Gaussian approximation of the entanglement entropy.
and assuming a constant transmission probability D, the logarithm of the moment generating function reads [27]
logχ(λ) = −tkT
h u+u−, (3.33)
where
u± =v ±Arcosh [Dcosh(v+iλ) + (1−D) cosh(v)],
and v = V /(2kT) is the ratio of the voltage over the temperature. For our tight-binding chain, the transmission probability is energy-dependent, see Eq. (2.36). To make a connection with Eq. (3.33), we consider the average transmissionD over the voltage window, Eq. (2.37). Since we apply a symmetric bias, V /2 = µL =−µR, the
whereθ=tQPC/¯t. Using this expression for the transmission probability, our results for the entanglement entropy in Fig. 3.6 are in excellent agreement with predictions based on Eq. (3.33).
From the ndings above we see that the entanglement entropy increases due to three types of processes: quantum noise at zero bias and at zero temperature, which gives rise to the logarithmic behavior in Eq. (3.29), together with thermal and shot
noise uctuations that cause a linear increase with time. In an experiment, the shot noise contribution can be suppressed simply by not applying a bias. However, thermal uctuations, which will dominate over the quantum noise at long times, will always be present. Thus, to access the logarithmic short-time behavior due to quantum noise, it has been suggested to open and close the QPC in a periodic manner [69].
To describe the periodic opening and closing of the QPC, we replace f(t) in Eq. (3.27) by the periodic function
F(t) = X∞ n=0
[f(t−nT)−f(t−w−nT)]. (3.35) Here T is the period of the driving and w is the length of the time window during which the QPC is open. Choosing the time window to be shorter than the thermal time scale, w < tβ, we expect to see a recurrence of the logarithmic behavior after each opening of the QPC. This is conrmed by our calculations shown in Fig. 3.6b.
Importantly, the exact results for the entanglement entropy are very well captured by the Gaussian approximation that only includes the second cumulant. Moreover, the second cumulant of the full counting statistics can be related to the current noise, thereby paving the way for an experimental verication of predictions from conformal eld theory in a coherent electronic conductor.